1. Field of The Invention
The present invention relates generally to a system for controlling a brushless DC motor, and more particularly, to a system capable of minimizing a torque ripple.
2. Description of Related Art
Various kinds of motors are used as driving devices for industrial electrical devices, such as a video tape recorder or a color printer, for example, and automatic machinery and tools, such as a robot, a numerical control machine, and the like. Demand for such motors is rapidly increasing because of automation.
Motors are broadly classified as DC motors and AC motors. DC motors are generally used because they are easily controlled. However, a disadvantage of the DC motor is the necessity for replacing brushes periodically. Although an AC motor does not require brushes, it is difficult to control. Thus, it has not been used for industrial products of high accuracy. However, because control technique for the AC motor are greatly advanced due to the development of microprocessor and other electronic techniques, the AC motor is replacing the DC motor.
Specifically, brushless DC motors and synchronous AC motors are generally used in highly accurate servomechanisms. The brushless DC motor is advantageous in that production costs are low. However, because brushless DC motors are difficult to control, they have been generally used for automatic machinery and tools, and industrial electrical devices which do not require micro-precision accuracy.
First, the operating principle of the brushless DC motor is described with reference to FIG. 1. Construction of a three-phase brushless DC motor is similar to that of a synchronous AC motor. A rotor 13 consists of a permanent magnet having P pole pairs (one pair being shown). Coils 11 are wound in a stator iron core 12, and three-phase current is supplied through the coils 11.
When the rotor 13 of the permanent magnet is rotating at a speed D, a magnetic flux is generated. The magnetic flux cuts current flowing through the coils 11 of the stator, thereby generating a torque between the stator iron core 12 and the rotor 13 according to Ampere's Law (F=I.times.B). Arrows illustrated in the rotor 13 of FIG. 1 indicate the directions of the magnetic flux.
Secondly, a dynamic equation of the brushless DC motor is derived based upon the following assumptions of a three-phase brushless DC motor.
1) The motor operates in a linear area and a the hysteresis loss may be ignored.
2) A slit between the rotor and the stator is uniform compared to the radius of the rotor.
3) Mutual inductances among each phase are constant regardless of the position of the rotor.
Since phase voltage of the motor is the sum of the amount of voltage drop caused by resistance and a time derivative of the amount of magnetic flux interlinkage, an equation of the phase voltage can be expressed as follows: ##EQU1##
where V.sub.k is the input voltage to the phase, i.sub.k is the phase current, r.sub.k is phase resistance and .lambda..sub.k is the magnetic flux.
The interlinked magnetic flux of one phase is classified into a magnetic flux produced by current flowing in the phase, a magnetic flux interlinked by current of the other phase and a magnetic flux interlinked by the rotor of the permanent magnet, and the motor is operated in the linear area (Assumption 1). The magnetic flux .lambda..sub.k can be obtained as follows; ##EQU2##
where L.sub.jk is a mutual inductance between phases j and k, L.sub.kk is a self-inductance, and .lambda..sub.mk is the interlinkage number of the magnetic flux between the rotor and a phase, k.
If the equation (2) is substituted into the equation (1), ##EQU3##
where .theta. is a position of the rotor.
The last term in the right-hand side of the equation (3) corresponds to the reverse electromotive force of the motor, and is the speed of the rotor multiplied by a position .theta. derivative of the interlinkage number of the magnetic flux interlinked by the rotor on the stator. Accordingly, most of voltage applied to the motor is cancelled by the reverse electromotive force, which has a large value at high speed, such that it is difficult to obtain current of a predetermined flow and thereby a desired torque can not be obtained.
The following may be defined according to condition (3). ##EQU4##
The position .theta. derivative of the interlinkage number of the magnetic flux interlinked by the rotor on the stator is a characteristic function of the motor, and since phases have phase differences by 2/3.pi. from one another, the following definitions may be obtained. ##EQU5##
Since three phases are connected in a Y line, the total of the current of the three phases becomes zero. EQU i.sub.1 +i.sub.2 +i.sub.3 =0 (6)
According to the conditions, (3), (4), (5) and (6), a following equation (7) may be obtained. ##EQU6##
The equation (7) can be expressed for the three phases as follows; ##EQU7##
A coenergy relationship is introduced to calculate the torque generated by one phase. ##EQU8##
Referring to equation (10), the torque generated by one phase is the characteristic function g(.theta.) of the motor multiplied by the phase current. Since the entire torque T is the total of T.sub.1, T.sub.2 and T.sub.3, an equation (11) may be obtained according to equations (5) and (10). EQU T=T.sub.1 +T.sub.2 +T.sub.3 =g(.theta.)i.sub.1 +g(.theta.-2/3.pi.)i.sub.2 +g(.theta.+2/3.pi.)i.sub.3 ( 11)
g(.theta.) in the equation (11) is the characteristic function, and may have a trapezoidal wave form or a sinusoidal wave form or a quasi-sinusoidal wave form. Accordingly, the user should obtain a suitable current command according to the wave form of g(.theta.) to generate the desired torque, and make the actual current follow a current command by adjusting the phase voltage since the actual current is controlled by a dynamic equation shown in the equation (8). Referring to the equation (11), the torque is expressed by the function of .theta., and also the current command should be the function of .theta. to obtain the torque free of .theta..
The dynamic equation for a mechanical part is expressed as the following equation (12), and J corresponds to a moment of intertia of the motor and B to a frictional coefficient of the motor. EQU Jd.omega./dt+B=T-T.sub.L .omega. (12)
Third, a mathematical model of the brushless DC motor based on the equations (8) and (11) is shown in FIG. 2.
Fourth, a method for controlling the torque of the conventional brushless DC motor--derivation of the current command for minimizing the torque ripple--is as follows.
The dynamic equation of the brushless DC motor is as follows: ##EQU9##
Referring to the equation (16), the torque is the function of .theta. as well as of the current. Since the term of g(.theta.) is the characteristic function of the motor, the phase current should be controlled to generate the desired torque.
To make the phase current directly proportional to the current, as in the conventional DC motor, the user should obtain the current command which makes the torque free of .theta. under the assumption that the actual current perfectly follows the current command according to the operation of a current controller.
An equation (17) obtains the current command, which makes the torque proportional to the current as follows: i.sub.M *(t) is a torque command produced from the speed controller, f.sub.k (.theta.) is a wave form of the current command which makes the torque free of .theta., that is, without a ripple. EQU i.sub.1 *(i.sub.M *,.theta.)=i.sub.M *(t)f.sub.1 (.theta.) EQU i.sub.2 *(i.sub.M *,.theta.)=i.sub.M *(t)f.sub.2 (.theta.) EQU i.sub.3 *(i.sub.M *,.theta.)=i.sub.M *(t)f.sub.3 (.theta.) (17)
Torque may be obtained by substituting the equation (17) into the equation (16) if the phase current governed by the equations (13), (14) and (15) perfectly follows the current command expressed in the equation (17) according to the operation of the current controller. EQU T(i.sub.M *,.theta.)=i.sub.M *[f.sub.1 (.theta.)g(.theta.)+f.sub.2 (.theta.)g(.theta.-2/3.pi.) +f.sub.3 (.theta.)g(.theta.+2/3.pi.)](18)
If the equation (19) is satisfied, the torque ripple .theta. does not exist since the torque is proportional to the current, and the torque is the function of only i.sub.M *. EQU f.sub.1 (.theta.)g(.theta.)+f.sub.2 (.theta.)g(.theta.-2/3.pi.)+f.sub.3 (.theta.)g(.theta.+2/3.pi.)]=k (19)
Then, f.sub.k (.theta.), satisfying the equation (19) for the g(.theta.), will be obtained. If the g(.theta.) has the trapezoidal wave form, it has a similar area to a linear area. When the current command is zero in the linear area and constant in a constant area, the torque ripple does not exist. ##STR1##
Lastly, f.sub.k (.theta.) satisfying the equation (19) may expressed as an equation (21), and have a spherical wave as shown in FIGS. 3A-3C. ##STR2##
Then, EQU f.sub.1 (.theta.)g(.theta.)+f.sub.2 (.theta.)g(.theta.-2/3.pi.)+f.sub.3 (.theta.)g(.theta.+2/3.pi.)]=2M (22)
When the actual current perfectly follows the current command of the spherical wave, as a result, the torque is made to be proportionate to i.sub.M *(t). EQU T=2Mi.sub.M *(t) (23)
A Korean Examined Application No. 93-4030 entitled "A Method For Improving A Torque Ripple Of A Brushless DC Motor" published on May 19, 1994, proposes overcoming the torque ripple of a brushless DC motor.
In the above publication a method for driving the linear voltage which allows a slight tilt at a point of turning ON/OFF, is proposed. The above-identified disclosure proposes to overcome the problems and disadvantages of a conventional method for driving a constant current, so that optimal current may flow to improve the torque ripple. This is carried out by controlling an electrothermosensitive angle and the tilt at the time of turning ON/OFF, namely, a slew rate of an active amplifier in an inverter terminal to maximize an average torque as much as possible after minimizing the current in the method for driving linear voltage and rotating a torque pulsating exponent to a minimum.
However, the above publication is for minimizing the harmonics of the output current, and the relationship between the output current and an output torque is not clearly disclosed.
Since the output torque is a multiplication of the reverse electromotive force and the output current, the ripple is not removed even though the harmonics of the output current is reduced. A certain restriction needs to be made to the wave form of the reverse electromotive force to remove the ripple.
In addition, in the above-identified publication, the method which controls the slew rate of the amplifier in the inverter terminal to apply the current command is approximate. A more exact method for controlling current is required.
A conventional circuit for controlling the torque and inverter is illustrated in FIG. 4.
Referring to FIGS. 5A-5C, current of iM*, -iM* flows in two phases at every range of .pi./6, and current does not flow in the third phase. Accordingly, it is preferable for current among each line to flow in the amount of iM* uniformly.
For example, the current between each line from phase a, to phase between 0 and .pi./6, is made to be i.sub.M *. The current command in the phase c, is zero, and it is preferable to modify a winding rather than to control the current command in the phase c.
Each unit in a torque control circuit and an inverter circuit has the following role.
1) A derotator circulates current of one phase among three phases according to the position of the rotor.
2) An error amplifier amplifies a current error signal.
3) A PWM amplifier converts the amplified error signal into a pulse width modulation signal.
4) A rotator supplies BUS voltage and -BUS voltage between each line of two phases to be controlled by the PWM signal, and breaks down the wire of the third phase.
5) A commutation logic receives information about position of the rotor and selects two phases to be controlled.
A high gain controller is used as the current controller, and FIG. 7 is a block diagram generally illustrating the system for controlling the brushless DC motor including the position and speed controllers.
A transfer function between i.sub.1 *, i.sub.1 will be gained as follows when the reverse electromotive force g(.theta.).omega. is ignored. ##EQU10##
Then, a bandwidth of a control loop becomes (R+K)/L. However, since the current command i.sub.1 * has the spherical wave form and has an infinite frequency component, K should be infinite in order that i.sub.1 follows i.sub.1 *. However, an infinite K is impossible. The actual current is shown in FIGS. 6A-6D, and the torque ripple is produced six times for each rotation due to the current control error.
Since the reverse electromotive force g(.theta.).omega. becomes greater and the voltage offset by input voltage v.sub.1 for controlling current is applied to the motor, the efficiency of controlling the current is reduced and the frequency of torque ripple becomes high when the motor is rotated at a high speed.
Since the torque ripple is produced six times for each rotation, it can be expanded in Fourier series as shown in the following equation (25). ##EQU11##
Since the frequency of the ripple becomes high and is filtered in a low pass filter, by inertia of the motor when rotated at a high speed, the problem is minor. However, since the frequency of the ripple is represented as a speed ripple, there is a problem in that the speed control is not exact when the motor is rotated at a low-speed.
Since the current command is a spherical wave, a discontinuous function, the actual current does not closely follow the command. Thus, the torque ripple is produced in the conventional torque controlling method.
In light of the foregoing, there is a need to provide a system for controlling a brushless DC motor that overcomes the problems and disadvantages of the conventional devices.